CatGloss

Let $\mathscr{A}$ be a category, $\mathbf{I}$ a small category, and $D{\colon}\linebreak[0] \mathbf{I} \to \mathscr{A}$ a diagram in $\mathscr{A}$. 1. A cone on $D$ is an object $A \in \mathscr{A}$ (the vertex of the cone) together with a family $$\Bigl( A \stackrel{f_I}{\longrightarrow} D(I) \Bigr)_{I \in \mathbf{I}}$$ of maps in $\mathscr{A}$ such that for all maps $I \stackrel{u}{\longrightarrow} J$ in $\mathbf{I}$, the triangle \(\xymatrix@R=1ex{ &D(I) \ar[dd]^{Du} \\ A \ar[ru]^{f_I} \ar[rd]_{f_J} & \\ &D(J) }\) commutes. (Here and later, we abbreviate $D(u)$ as $Du$.)\n2. A limit of $D$ is a cone $\Bigl(L \stackrel{p_I}{\longrightarrow} D(I)\Bigr)_{I \in \mathbf{I}}$ with the property that for any cone~ on $D$, there exists a unique map $\bar{f}{\colon}\linebreak[0] A \to L$ such that $p_I \circ \bar{f} = f_I$ for all $I \in \mathbf{I}$. The maps $p_I$ are called the projections of the limit.

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