[[Category theory MOC]]
# Arrow category
The **arrow category** $\cat C^{\to}$ of a [[category]] $\cat C$ has morphisms of $\cat C$ as objects,
and for $f \in \cat C(X,Y)$ and $f' \in \cat C(X',Y')$
a morphism $g \in \cat C^{\to}(f,f')$ is a pair of morphisms $g_{1} \in \cat C(X,X')$ and $g_{2} \in \cat C(Y,Y')$ such that the following diagram commutes:[^2010] #m/def/cat 

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[^2010]: 2010\. [[Sources/@awodeyCategoryTheory2010|Category theory]], p. 15


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