When the canonical comparison of the pushout induced by X(A,-) is injective? This is my first time asking here. I would like to understand the following:
Given a morphism $p : E\to B $ and any object $ A$ of a category $\mathbb{A} $, consider the pushout of $ p $ along $ p $ and the pushout of $\mathbb{A} (A, p ) $ along itself. We denote them respectively by $P$ and $P' $. 
There is a canonical comparison $ P' \to \mathbb{A} (A, P ) $. 
When this comparison is injective? 
If I am not mistaken, this is true for $ \mathbb{Set } $.  Is there any exact condition that implies in this condition? 
Is there any counterexample?
Thank you in advance
 A: If $p$ is an epimorphism, then its pushout along itself $P$ is given (up to isomorphism) by
$$\require{AMScd}\begin{CD}E@>{p}>>B \\ @V{p}VV@VV{id_B}V \\ B @>>{id_B}> B,
\end{CD}$$
and this property characterise epimorphisms. Such a commutative diagram is preserved by any functor, so in particular taking the image by $\mathbb{A}(A,\_)$ would give you
$$\begin{CD}\mathbb{A}(A,E) @>{\mathbb{A}(A,p)}>> \mathbb{A}(A,B) \\ @V{\mathbb{A}(A,p)}VV@VV{id_{\mathbb{A}(A,B)}}V \\ \mathbb{A}(A,B) @>>{id_{\mathbb{A}(A,B)}}> \mathbb{A}(A,B).
\end{CD}$$
In particular, the comparison $P'\to \mathbb{A}(A,B)$ would be a split epimorphism, so that if it is a mono, it is an isomorphism, and then the two canonical maps $\mathbb{A}(A,B)\to P'$ are also isomorphism. This would imply that $\mathbb{A}(A,p)$ is also an epimorphism, and in particular, that it would be surjective. In the specific case where $A=B$, this would mean that there exist $s\in \mathbb{A}(B,E)$ such that $p\circ s=\mathbb{A}(A,p)(s)=id_B$, i.e. that $p$ must be a split epimorphism.
Thus any epimorphism that isn't split in $\mathbb{A}$ will give you a counterexample.
