I claim that a such category exist:
Let $\mathcal{C}$ any category, build a full immersion $\mathcal{C}\subset \mathcal{C'} $ adding to $\mathcal{C}$ the new objects: $\coprod_f d_0(f), \ \coprod_f d_1(f) $ and the no-identity new arrows: $\coprod_f f: \coprod_f d_0(f) \to \coprod_f d_1(f)$ and for any $f: X\to Y$ the arrows $\epsilon^0_f: X\to \coprod_f d_0(f)$ (think as a f-coprojection of $X=d_0(f)$) and $\epsilon^1_f: X\to \coprod_f d_1(f)$.
Then the $\mathcal{C'}$ arrows are the $\mathcal{C}$ arrow more the following:

$\coprod_ff\star \epsilon^0_g\star h $ , $ \star \epsilon^0_g\star h $, $ \epsilon^1_g\star h $ ($h\in \mathcal{C}\downarrow d_0(g) $)
with obvious composition (here "$\star$" is a "free" composition, and composing morphisms of $\mathcal{C}$ by original compositionlaw " $\circ$ " whenever possible).

Then in $\mathcal{C'}$ we consider the congruence : $\coprod_ff\star \epsilon^0_f\star g \sim \epsilon^1_f\star (f\circ g) $

And let $\mathcal{C''}$ the quotient category, still we have a full immersion $\mathcal{C}\subset \mathcal{C''} $

And in $\mathcal{C''}$ we have the commutative diagram:

1] \xymatrix{X\ar[r]^{\epsilon^0_f }\ar[d]_f &\coprod_fd_0(f) \ar[d]^{\coprod_ff}\Y\ar[r]_{\epsilon^1_f}&\coprod_fd_1(f)}
i.e. $\coprod_ff\circ \epsilon^0_f = \epsilon^1_f \circ f $

but the follow isn't commutative (commutative only if $f=g=h$):

2] \xymatrix{X\ar[r]^{\epsilon^0_g }\ar[d]_h &\coprod_fd_0(f) \ar[d]^{\coprod_ff}\Y\ar[r]_{\epsilon^1_f}&\coprod_fd_1(f)}
i.e. $\coprod_ff\circ \epsilon^0_g \neq \epsilon^1_f \circ h $

Then [1] is a Pullback, and $\coprod_f f: \coprod_f d_0(f) \to \coprod_f d_1(f)$ classifing any arrow of $\mathcal{C''} $

(sorry for diagrams, but I hope what I mean is understable)