Definition 1   Let $\mathcal{C}$ be a category. A diagram in $\CC$ is a directed graph $\GG$ with vertex set $V$ and edge set $E$ , (``loops'' and ``parallel edges'' are allowed) together with two maps $o\co V\to\mathrm{Obj}(\CC)$ , $m\co E\to \mathrm{Morph}(\CC)$ such that if $e\in E$ has source $s(e)\in V$ and target $t(e)\in V$ then $m(e) \in {Hom}_{\CC}\left(o\left(s(e)\right),o\left(t(e)\right)\right)$ .

Definition 2   Let $D=(\GG,o,m)$ be a diagram in the category $\CC$ and $\Gg=(e_1,\ldots,e_n)$ be a path in $\GG$ . Then the composition along $\Gg$ is the following morphism of $\CC$ $$\circ(\Gg):=m(e_n)\circ\cdots\circ m(e_1)\,.$$ We say that $D$ is commutative or that it commutes if for any two objects in the image of $o$ , say $A=o(v_1)$ and $B=o(v_2)$ , and any two paths $\Gg_1$ and $\Gg_2$ that connect $v_1$ to $v_2$ we have $$\circ(\Gg_1)=\circ(\Gg_2)\,.$$