This question may be "evil", if so I apologize in advance.
$\newcommand{\obj}{\operatorname{Obj}} \newcommand{\category}{\mathscr{C}} \newcommand{\mor}{\operatorname{Mor}}$ Let's say one has a category $\category$ and a sub-category $\category'$, where $\obj(\category)=\obj(\category')$ but for all objects $C_1, C_2 \in \obj(\category)=\obj(\category')$, one has that $\mor_{\category'}(C_1, C_2) \subset \mor_{\category}(C_1,C_2)$. (In particular, for any object $C_1$, one has that $\mor_{\category'}(C_1,C_1)$ is missing "many" automorphisms that are found in $\mor_{\category}(C_1,C_1)$, which may make this question "evil".)
However, one has that for every morphism $f$ in $\mor_{\category}(C_1,C_2)$, it can be factorized as: $$ f = \mathscr{A}_2 \circ f' \circ \mathscr{A}_1 $$ where $f' \in \mor_{\category'}(C_1,C_2)$, $\mathscr{A}_1$ is an automorphism in $\category$ of $C_1$ possibly with $\mathscr{A}_1 \not\in \mor_{\category'}(C_1,C_1)$, and $\mathscr{A}_2$ is an automorphism in $\category$ of $C_2$ possibly with $\mathscr{A}_2 \not\in \mor_{\category'}(C_2,C_2)$.
In other words, all morphisms in $\category$ can be generated from this special subclass of morphisms using automorphisms.
Question: What is the name for this special type of subcategory, or this special class of morphisms?
To clarify, the example I have in mind is where $\category$ is the sub-category of finite vector spaces consisting only of $\mathbb{R}^n$ for some $n$, and $\category'$ has its morphisms restricted to linear functions which send the standard basis of $\mathbb{R}^{n_1}$ to some subset of the standard basis of $\mathbb{R}^{n_2}$ or $\vec{0}$ (I believe these are closed under composition). Then all other morphisms of $\category$, i.e. linear functions $\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$, can be generated by using "change of basises" as the automorphisms.