Slice categories, composition, and pullback along a morphism

For an object $X$ of a category $\mathcal{C}$, the slice category $\mathcal{C}\mathrm{/}X$ has objects arrows $a:A\to X$ and morphisms commuting triangles, as in Comma category, slice category, and coslice category.

For a morphism $f:X\to Y$, postcomposition with $f$ defines a functor

${\mathrm{\Sigma }}_f:\mathcal{C}\mathrm{/}X\to \mathcal{C}\mathrm{/}Y,(a:A\to X)⟼(f\circ a:A\to Y)\mathrm{.}$

If chosen pullbacks along $f$ are supplied, they define the pullback functor

$f^{\ast }:\mathcal{C}\mathrm{/}Y\to \mathcal{C}\mathrm{/}X,$

sending $b:B\to Y$ to a chosen pullback square

$\begin{array}{c}X{\times }_{Y}B\to B\\ \downarrow \downarrow b\\ X\stackrel{f}{\to }Y\end{array}$

and sending a morphism in $\mathcal{C}\mathrm{/}Y$ to the unique induced morphism between the chosen pullbacks (Pullbacks and pushouts as limits and colimits of cospans and spans). For composable base maps $X\stackrel{f}{\to }Y\stackrel{g}{\to }Z$, the universal property supplies a canonical natural isomorphism

$f^{\ast }{g}^{\ast }\cong (gf{)}^{\ast }\mathrm{.}$

Likewise $(1_X{)}^{\ast }\cong 1_{\mathcal{C}\mathrm{/}X}$. Thus arbitrary pullback choices are contravariantly pseudofunctorial in the base. Literal functorial equalities in the sense of Covariant functor, identity functor, composite functor, and contravariant functor require coherent split choices and do not follow merely from choosing each pullback separately.