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Example 4.31.3. Let $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ be categories. Let $F : \mathcal{A} \to \mathcal{C}$ and $G : \mathcal{B} \to \mathcal{C}$ be functors. We define a category $\mathcal{A} \times _\mathcal {C} \mathcal{B}$ as follows:

  1. an object of $\mathcal{A} \times _\mathcal {C} \mathcal{B}$ is a triple $(A, B, f)$, where $A\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$, $B\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$, and $f : F(A) \to G(B)$ is an isomorphism in $\mathcal{C}$,

  2. a morphism $(A, B, f) \to (A', B', f')$ is given by a pair $(a, b)$, where $a : A \to A'$ is a morphism in $\mathcal{A}$, and $b : B \to B'$ is a morphism in $\mathcal{B}$ such that the diagram

    \[ \xymatrix{ F(A) \ar[r]^ f \ar[d]^{F(a)} & G(B) \ar[d]^{G(b)} \\ F(A') \ar[r]^{f'} & G(B') } \]

    is commutative.

Moreover, we define functors $p : \mathcal{A} \times _\mathcal {C}\mathcal{B} \to \mathcal{A}$ and $q : \mathcal{A} \times _\mathcal {C}\mathcal{B} \to \mathcal{B}$ by setting

\[ p(A, B, f) = A, \quad q(A, B, f) = B, \]

in other words, these are the forgetful functors. We define a transformation of functors $\psi : F \circ p \to G \circ q$. On the object $\xi = (A, B, f)$ it is given by $\psi _\xi = f : F(p(\xi )) = F(A) \to G(B) = G(q(\xi ))$.

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