The question cannot be completely wrong, as it does not completely make sense, whence let me detail a bit. I'm in a monoidal category $(\mathcal{C},\otimes,\mathbb{I},\alpha,\lambda,\rho)$ with pushouts and pullbacks and such that $X\otimes-$ and $-\otimes X$ preserve pushouts for every object $X$. I have a diagram of the form $$ \begin{array}{ccccc} e & \overset{\eta}{\rightarrow} & a & \underset{g}{\overset{f}{\rightrightarrows}} & b \\ & & p \downarrow & & \downarrow q \\ e' & \underset{\eta'}{\rightarrow} & c & \underset{k}{\overset{h}{\rightrightarrows}} & d \end{array} $$ where $qf = hp$, $qg = kp$, both $p$ and $q$ are epis, $(e,\eta$), $(e',\eta')$ are equalizers and there exists $\sigma:b\to a$, $\sigma':d\to c$ such that $\sigma f = 1_a = \sigma g$ and $\sigma'h=1_c=\sigma'k$. By universal property, there exists a unique morphism $\pi:e\to e'$ such that $\eta'\pi=p\eta$. I'm interested in two situations:
Question 1: When can I claim that $\pi$ is epi as well?
Question 2: When is $$ \begin{array}{ccc} e & \overset{\eta}{\rightarrow} & a \\ \pi \downarrow & & \downarrow p \\ e' & \underset{\eta'}{\rightarrow} & c \end{array} $$ a pushout square?
In the very particular case I started to work with, $\mathcal{C}$ was the category of vector spaces over a field and, by computing directly with elements and diagram chasing, it happened that the map induced by $f-g$ between $\ker p$ and $\ker q$ was surjective, whence $\pi$ had to be surjective as well by snake lemma and the square became a pullback, whence a pushout. However, now I would like to extend further this result. Again by direct check it works over the opposite category of sets, with the unique difference that the square is directly and only a pushout and not a pullback. Is it, by chance, that there is some deep categorical reason for these to hold?
From this my general question:
Question 0: When do the canonical map induced at the level of equalizers by two epis is still an epi?