CatGloss

Any map $A \stackrel{f}{\longrightarrow} A’$ in $\mathscr{A}$ induces a natural transformation $ \xymatrix@C+1em{ \mathscr{A}^\op \rtwocell<4>^{\h_A}{\h{A’}}{\hspace{.5em}\h_f} &\mathbf{Set} } $(also called$ \mathscr{A}( - , f) $,$ f_* $or$ f \circ - $), whose component at an object$ B \in \mathscr{A} $is$ \begin{array}{ccc} \h_A(B) = \mathscr{A}(B, A) &\to & \h_{A’}(B) = \mathscr{A}(B, A’) \ p &\mapsto & f \circ p. \end{array} $Let$ \mathscr{A} $ be a locally small category. The Yoneda embedding of $\mathscr{A}$ is the functor $ \h_\bullet{\colon}\linebreak[0] \mathscr{A} \to \ftrcat{\mathscr{A}^\op}{\mathbf{Set}} $ defined on objects $A$ by $ \h_\bullet(A) = \h_A $ and on maps $f$ by $ \h_\bullet(f) = \h_f $ . Here is a summary of the definitions so far.

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