I am studying the theory of $\infty-$category. I have seen the construction of pullbacks in $\infty-$category. I want to know whether this property can be checked like in simplical category. Do we have $Map(colim(A_i),B) \simeq lim(Map(A_i,B))$ or $Map(A,lim(B_i)) \simeq lim(Map(A,B_i))$ just as in simplical category or usual category?
I have checked HTT and Kerodon. But I only found the results when the $\infty-$category is the homotopy coherent nerve of some locally Kan simplical category. I don't know whether it is enough.