I encountered the titular statement in a discussion on subobject classifiers (in particular: in the proof that the domain of such a subobject classifier is terminal).
However, I think his 'basically' does a little too much heavy lifting here. Consider a category with pullbacks.
Consider the diagram $A \xrightarrow{m} T \xleftarrow{\text{id}_T} T$. Then it has some pullback $A \xleftarrow{g} A \times_T T \xrightarrow{f} T$.
Is it really true that $g$ is somehow the identity...? I mean, its source and target prevent that already. After a lot of fiddling around with the universal property of pullbacks (choosing arbitrary objects and maps to $A$ and $T$, of specific ones meeting the pullback requirement), I have not come up with a satisfying answer.
So what's going on here? Is $g$ 'basically' the identity in the sense that precomposing with the unique map $h : A \to A \times_T T$ yields it? i.e., is $g$ the right-inverse to the unique map provided by the universal property of the pullback?