I'm interested in the following construction of inverse system of groups. Let $((X_i)_{i\in I},(f_{ij})_{i\le j\in I})$ be an inverse system over $I$ and $\phi:J\rightarrow I$ be a morphism between posets. We can define an inverse system $((Y_i)_{i\in J},(g_{ij})_{i\le j\in J})$ over $J$ as $Y_i:=X_{\phi(i)}$ and $g_{ij}:=f_{\phi(i)\phi(j)}$.
Moreover, let $(X,(p_i)_{i\in I})=\lim_{i\in I}(X_i)$ and $(Y,(q_i)_{i\in J})=\lim_{i\in J} (Y_i)$. For $i\in J$, define $\psi_i:=p_{\phi(i)}:X\rightarrow Y_i$. Then $(\psi_i)$ induces a morphism $\psi:X\rightarrow Y$ such that $q_i\circ\psi=\psi_i$ for $i\in J$.
Question: Does this construction have a name (maybe "pullback"?) and will $\psi$ be injective, surjective or bijective if $\phi$ is injective, surjective or bijective respectively?
I know there are many analogous constructions in mathematics, for example, pullback of presheaves. But I didn't find any references mentioned this construction concerning inverse system of groups. I would be grateful if you could provide a reference.
If your inverse system is represented as a functor $X : (I,\ge) = (I,\le)^{op}\to C$ (what's your codomain category by the way? Where do the $X_i$ belong?) then you can define an inverse system $\phi^*X = X \circ \phi^{op} : J^{op}\xrightarrow{\phi^{op}} I^{op} \xrightarrow{X} C$ as composition of functors.
This is a convenient way to see the construction, because (for example) the theory of final functors will give you sufficient conditions so that $\lim_i X_i\cong \lim_j (\phi^* X)_j$.