Assume we are working in a (non-degenerate) Cartesian closed category $\mathbf{C}$:
Lets say we have arrows $f,\tilde{f}:x\rightrightarrows a$ and $g:y\to a$ such that the square $$\begin{array}{c c c} P & \xrightarrow{p} & x\\ \downarrow q & & \downarrow \\ y & \xrightarrow{g} & a\end{array}$$ is a pullback when the $x\to a$ arrow is $f$ and also when it is $\tilde{f}$.
Assume that $p$ (the pullback of $g$ along $f$ or $\tilde{f}$) is monic.
Assume also that, for any third object $z$ and arrows $h:z\to x$, $k:z\to y$ the unique arrow $z\to P$ making everything conmute is the same regardless of the $x\to a$ arrow we use.
Can we conclude that $f=\tilde{f}?$
If this is not true in general, is it true if $y$ is a terminal object?
(In the slice category $\mathbf{C}/B$, I'm trying to prove that $\langle\chi_f,v\rangle$ is the characteristic arrow of a monic $f$ from $(u:X\to B)$ to $(v:Y\to B)$ if that helps)
Thanks!
It need not be true; for example in any pointed category, all monomorphisms have trivial kernel, so any square $$\require{AMScd} \begin{CD} 0 @>>> X \\ @VVV @VV{m}V\\ 0 @>>> Y \end{CD}$$ where $m$ is a monomorphism is a pullback, but you can have plenty of different monomorphisms $m\colon X\to Y$.
Note that such examples exists in categories of modules, groups, Lie algebras, monoids, pointed sets... and that limits in these categories are preserved by the corresponding "underlying set" functors, which also gives you plenty of examples in the category of sets.
For another counterexample in the category of sets (or any topos), just to take any function $f\colon X\to Y$, then the square $$\begin{CD} \{*\} @>>> X\sqcup\{*\} \\ @VVV @VV{f\sqcup 1}V\\ \{*\} @>>> Y\sqcup \{*\} \end{CD}$$ is a pullback.