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I've just started to read about Category Theory.

More precisely nLab, opposite category. I'm trying to understand the expression: $$ \circ_{C^{op}} : C_{\mathrm{mor}}^{op} {}_{s^{op}}\times_{t^{op}} C_{\mathrm{mor}}^{op} = C_{\mathrm{mor}} {}_{t} \times_{s} C_{\mathrm{mor}} \stackrel{\simeq}{\to} C_{\mathrm{mor}} {}_{s} \times_{t} C_{\mathrm{mor}} \stackrel{\circ}{\to} C_{\mathrm{mor}} \,, $$

Are the indices of $\times$ something that can be explained within the set theoretical Cartesian product or it is something strictly connected to Category Theory (it seems to occur near the word pullback)? In the first case I would appreciate a short explanation, in the second case I'll wait to get there.

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As you say, these indices are used to denote a pullback rather than a (cartesian) product. In a general category, given two arrows with the same codomain $X\stackrel{f}\longrightarrow Z\stackrel{g}\longleftarrow Y$, their pullback is an object, denoted $X\ {_f\times_g }\ Y$ (or more frequently, $X\times_Z Y$), and equipped with two maps $\pi_1 :X\ {_f\times_g }\ Y\to X$ and $\pi_2:X\ {_f\times_g }\ Y\to Y$ such that $f\circ\pi_1=g\circ\pi_2$, wich is universal among such objects; this means that for every object $W$ and arrows $u:W\to X$ and $v:W\to Y$ such that $f\circ u=g\circ v$, there exist a unique map $w:W\to X\ {_f\times_g }\ Y$ such that $\pi_1 \circ w=u$ and $\pi_2\circ w=v$.

By contrast, the product of $X$ and $Y$ is defined by the same data of an object with two maps, but without the condition $f\circ\pi_1=g\circ\pi_2$, and it has the same property, but without the conditions $f\circ u=g\circ v$; so the pullback is a bit like a product but with an additional restriction. Incidentally, the pullback is often called "fibered product".

In the set-theoretical case, the pullback can be easily obtained from the product: whereas the product is the set of pairs $(x,y)$ such that $x\in X$ and $y\in Y$, the pullbacks is the set of such pairs with the additional condition that $f(x)=g(y)$. The two maps $\pi_1$ and $\pi_2$ are then just the restriction of the product projections to this set (i.e. $\pi_1(x,y)=x$ and $\pi_2(x,y)=y$).

In this specific case, the pullback $C_{\mathrm{mor}}\ {_{s} \times_{t}}\ C_{\mathrm{mor}}$ is the set of pairs of arrows $(\alpha,\beta)$ such that the source/domain of $\alpha$ is the target/codomain of $\beta$; in other words, it is the set of pairs of arrows that can be composed in your category $C$.

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