Cone over a diagram

A cone over $D:\mathcal{J}\to \mathcal{C}$ with vertex $A$ is a natural family ${\gamma }_j:A\to D(j)$ satisfying $D(u){\gamma }_i={\gamma }_j$ for every $u:i\to j$. Cone morphisms are maps of vertices commuting with every leg.

A limit of a diagram $D:\mathcal{J}\to \mathcal{C}$ is a terminal cone over $D$: every other cone factors through it uniquely.

A finite limit is a categorical limit whose indexing category has finitely many objects and morphisms.

A product of objects $(A_i)$ is a universal object equipped with projections to every $A_i$.

The equalizer of parallel arrows $f,g:A\rightrightarrows B$ is a universal arrow $e:E\to A$ satisfying $fe=ge$.

The pullback of $X\to Z\leftarrow Y$ is a universal commutative square with an object mapping to $X$ and $Y$. In the Category of sets, it is the set of pairs with equal images in $Z$.

If all small products and equalizers exist, the limit of $D:\mathcal{J}\to \mathcal{C}$ is the equalizer of the two mapswhose $u$-coordinates are respectively $D(u)$ after projection to $D(i)$ and direct projection to $D(j)$.

A category is complete when it has every small limit. Small products and equalizers suffice to construct all small limits.

A functor $F:\mathcal{I}\to \mathcal{J}$ is initial when every comma category $(F\downarrow j)$ is nonempty and connected. Restriction along an initial functor preserves limits:

For an initial functor $F:\mathcal{I}\to \mathcal{J}$, restriction gives an isomorphism between the category of cones over $D:\mathcal{J}\to \mathcal{C}$ and that over $DF$. Given a cone over $DF$, choose $(i,u:Fi\to j)$ and define its $j$-leg as $D(u){\gamma }_i$; connectedness of $(F\downarrow j)$ makes the result independent of the choice.