Let me first state that I am a category theory novice so your patience is appreciated. I might be making some very basic conceptional mistakes and I might just need a simpler language to do what I want to do (in particular, I seem to get down to the level of relations in objects..).
What I thought I was doing was defining a category as follows.
Take a category $\mathcal{C}_1$ made out of objects which all satisfy the axioms $\mathcal{I}$. Take a particular functor $F$ from $\mathcal{C}_1$ to $\mathcal{C}_2$. The 'image' of $\mathcal{C}_1$ under $F$ contains objects which satisfy "$F(\mathcal{I})$" but $F$ is not necessarily full. (Also there might be objects in $\mathcal{C}_2$ which satisfy "$F(\mathcal{I})$" but are not the image of any object from $\mathcal{C}_1$.
Therefore there would be this subcategory $\mathcal{C}_{F(\mathcal{I})}$ of $\mathcal{C}_2$ with objects defined as those objects satisfying $F(\mathcal{I})$ and morphisms the same as those of $\mathcal{C}_2$.
Is this an established notion or not a part of category theory (perhaps because of the reference to axioms of objects)?
Let me expand on my question by way of an example.
Consider the category $\mathbf{FinGrp}$ of finite groups with morphisms given as homomorphisms and the category $\mathbf{FinAlg_{\mathbb{C}}}$ of finite dimensional complex associative algebras with linear algebra homomorphisms for morphisms.
There is a (covariant) functor $\mathbb{C}:\mathbf{FinGrp}\rightarrow \textbf{FinAlg}_{\mathbb{C}}$ which associates to each group $G$ its group ring $\mathbb{C}G$ and to each group homomorphism $\varphi:G_1\rightarrow G_2$ the linear algebra homomorphism:
$$(\mathbb{C}\varphi):\mathbb{C}G_1\rightarrow \mathbb{C}G_2\,,\,\,\,\delta^g\mapsto \delta^{\varphi(g)}.$$
(is this OK?)
Now if we delve deeper into the objects of the category $\mathbf{FinGrp}$ we see some relations that hold for each object. Let $G$ be a finite group. There is an (associative) multiplication $m:G\times G\rightarrow G$, a unit $e\in G$ as well as an inverse ${}^{-1}:G\rightarrow G$.
The associativity of $m$ can presented as:
$$m\circ (I_G\times m)=m\circ (m\times I_G).$$
Define maps $L_e:G\rightarrow G\times G$ by $g\mapsto (e,g)$ and $R_e:G\rightarrow G\times G$ by $g\mapsto (g,e)$. Then the unit is axiomised as
$$m\circ L_e=I_G=m\circ R_e.$$
Finally inverses. Define $D:G\times G\rightarrow G$ by $g\mapsto (g,g)$ and $\pi_e:G\rightarrow G$ by $g\mapsto e$. Then inverses are axiomised by
$$m\circ({}^{-1}\times I_G)\circ D=\pi_e=m\circ(I_G\times{}^{-1})\circ D.$$
Let us call these three relations/predicates/axioms by $\mathcal{I}_a,\,\mathcal{I}_u$ and $\mathcal{I}_i$.
Does it make any sense or is there a canonical way to talk about the image of (these) predicates under a functor (for example the functor $\mathbb{C})$?
For example, the axioms of the group $G$ can be encoded in $\mathbb{C} G$ in the following way. I don't think this is anyway canonical: how would I describe this 'routine'? Should I just say that the group axioms look like or are translated in this way?
By the natural embedding $\iota:G\hookrightarrow\mathbb{C} G$, $g\mapsto \delta^g$, the group law on $G$ may be extended to a linear multiplication $\mathbb{C} G\otimes\mathbb{C} G\rightarrow \mathbb{C} G$. Start with the group law, a multiplication $m:G\times G\rightarrow G$ defined by $$m(g,h)=g\cdot h.$$ The multiplication on $G$ can be extended to a bilinear mapping $\overline{m}$ on $\mathbb{C} G\times \mathbb{C} G$ and via the universal property to a linear map $$\nabla: \mathbb{C} G\otimes \mathbb{C} G\rightarrow \mathbb{C} G.$$
The vector space $\mathbb{C} G$ together with the multiplication $\nabla$ is a complex associative algebra called the group ring of $G$.
$$\nabla\circ(\nabla\otimes I_{\mathbb{C} G})(\delta^g\otimes \delta ^h\otimes \delta^k)=\nabla\circ(I_{\mathbb{C} G}\otimes\nabla)(\delta^g\otimes \delta ^h\otimes \delta^k).$$ The identity element is encoded via the unit map $$\eta:\mathbb{C}\rightarrow\mathbb{C} G\,,\,\,\,\lambda\mapsto \lambda\delta^e.$$ Note that $\delta^e$ is the unit for the algebra $\mathbb{C} G$.
Note that using the isomorphisms $\mathbb{C}\otimes\mathbb{C} G\cong \mathbb{C}G\cong \mathbb{C} G\otimes \mathbb{C}$: $$\nabla\circ (\eta\otimes I_{\mathbb{C} G})(\lambda\otimes \delta^g)\cong I_{\mathbb{C}G}(\lambda\delta^g)\cong\nabla\circ(I_{\mathbb{C} G}\otimes\eta)(\delta^g\otimes \lambda).$$ To complete this process inverse elements must be considered. The inversion map $G\rightarrow G$, $g\mapsto g^{-1}$ may be extended to $\operatorname{inv}:\mathbb{C} G\rightarrow \mathbb{C} G$. Consider also the maps $\delta^g\mapsto \delta^g\otimes \delta^g$ and $\lambda \delta^g\mapsto \lambda$. Denote these maps by $D$ and $\pi_{\mathbb{C}}$ respectively. Now inverse elements are encoded by: $$\nabla\circ(\operatorname{inv}\otimes I_{\mathbb{C} G})\circ D(\lambda\delta^g)=\eta\circ \pi_{\mathbb{C}}(\lambda \delta^g)=\nabla\circ( I_{\mathbb{C} G}\otimes\operatorname{inv})\circ D(\lambda\delta^g).$$ Note that all of these maps are linear so it suffices to consider their action on basis elements only and so the group axioms are equivalent to the following relations between maps related to $\mathbb{C} G$:
\begin{eqnarray} \nabla\circ(\nabla\otimes I_{\mathbb{C} G})=\nabla\circ(I_{\mathbb{C} G}\otimes\nabla) \\ \nabla\circ(\eta\otimes I_{\mathbb{C} G})\cong I_{\mathbb{C} G}\cong \nabla\circ(I_{\mathbb{C} G}\otimes\eta) \\ \nabla\circ(\operatorname{inv}\otimes I_{\mathbb{C} G})\circ D=\eta\circ\pi_{\mathbb{C}}=\nabla\circ(I_{\mathbb{C} G}\otimes\operatorname{inv})\circ D. \end{eqnarray}
Is there any sense in calling these axioms by $\mathbb{C}(\mathcal{I}_a,\mathcal{I}_u,\mathcal{I}_i)$? Or referring to the subcategory of $\mathbf{FinAlg}_{\mathbb{C}}$ which satisfies $\mathbb{C}(\mathcal{I}_a,\mathcal{I}_u,\mathcal{I}_i)$ by $\mathcal{C}_{\mathbb{C}(\mathcal{I})}$ where $\mathcal{I}$ are these group axioms.
I interpret your question about group algebras to really be something like the following:
The answer is that group algebras have the additional structure of a Hopf algebra. The map $\text{inv}$ is part of this structure; it is called the antipode. The map $D$ is also part of this structure; it is called the comultiplication. All of the axioms for groups generalize to corresponding axioms for Hopf algebras. Moreover, the functor from groups to Hopf algebras is fully faithful and so restricts to an equivalence of categories.
But an arbitrary algebra doesn't come equipped with any $\text{inv}$ or $D$ maps, so it's unclear what it would mean to impose axioms on such an algebra involving them.