When $ X $ is a set, $ X^I $ is the set of functions from $ I $ to $ X $ , also written as $ \mathbf{Set}(I, X) $ . \minihead{Equalizers} To define our second type of limit, we need a preliminary piece of terminology: a fork in a category consists of objects and maps $ \xymatrix{ A \ar[r]^f & X \ar@<.5ex>[r]^s \ar@< - .5ex>[r]t & Y } $ such that $ sf = tf $ .Let $ \mathscr{A} $ be a category and let $ \parpairi{X}{Y}{s}{t} $ be objects and maps in $ \mathscr{A} $ . An equalizer of $ s $ and $ t $ is an object $ E $ together with a map $ E \stackrel{i}{\longrightarrow} X $ such that $ \xymatrix{ E \ar[r]^i & X \ar@<.5ex>[r]^s \ar@< - .5ex>[r]_t & Y } $ is a fork, and with the property that for any fork X, there exists a unique map $ \bar{f}{\colon}\linebreak[0] A \to E $ such that $ \begin{array}{c} \xymatrix{ A \ar@{.>}[d]{\bar{f}} \ar[rd]^f & \ E \ar[r]_i &X } \end{array} $ commutes.