HTT, 4.2.4.3, Diagonal embedding. This is Lem. 4.2.4.3 of HTT. > $K$ is a simplicial set. $C$ an $\infty$-category. There is a diagonal embedding $\delta:C \rightarrow Fun(K, C)$ What is this map and how does it exist? Here Fun(-,-) is the internal hom in $Set_\Delta$, category of simplicial sets. * * * The only possible map I can think of is the adjunct map induced by projection $$C \times K \rightarrow C $$

It's the map that takes $c$ to the constant functor whose value is $c$. As you guessed, this map is the adjunct to the projection $C \times K \to C$.