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Let $f:A\to C\leftarrow B:g$ be morphisms in a category. There exists in literature a useful notation for the morphisms $\bar f:A\times_C B\to B$ and $\bar g:A\times_C B\to A$ in terms of $f$ and $g$?

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Sometimes these morphisms are called projections, and denoted $p_A$ and $p_B.$ This is motivated by the example of the category of sets and set maps where we can describe the pullback as follows:

$$A\times_CB=\{(a,b)\mid f(a)=g(b)\}$$

Here the map $A\times_CB\to A$ takes $(a,b)$ to $a$, and $A\times_CB\to B$ takes $(a,b)$ to $b$, so in fact the two maps are projections.

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Yes, I would use this notation, or $\pi_A$ and $\pi_B$. – Qiaochu Yuan Jun 14 '13 at 5:33
These notations avoid the morphisms $f,g$ on which the pullback depends. I'm looking for a notation of the form $f\ltimes g:A\times_C B\to A$ and $f\rtimes g:A\times_C B\to B$, for example. – Fabio Lucchini Jun 14 '13 at 11:11
@user54738 The notation $A \times_C B$ also avoid $f,g$ on which it depends. It would be incoherent to mention $f,g$ in the notation of the projections and not on the notation of their domain. – Pece Jun 14 '13 at 15:05

Well, $A \times_C B \to B$ is the pullback of $f$ along $g$. Therefore it is often denoted as $f^*$ or (especially in algebraic geometry) as $f_B$.

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