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 $C \times K \to C$.