$\Delta\times p_0'$ sends any diagram $p'\in  {\operatorname}{Fun}(K,{{\mathscr{C}\mathrm{at}}}_\infty)$ to the $\infty$-category ${\operatorname}{Nat}_{K}(p_0',p')\in {{\mathscr{C}\mathrm{at}}}_\infty$ of natural transformations between $p_0'$ and $p'$. Since in the special case that $p_0'=\Delta_\ast$ is the constant diagram at the terminal $\infty$-category[^5] both functors and coincide, in particular, for any diagram $p'\in {\operatorname}{Fun}(K,{{\mathscr{C}\mathrm{at}}}_\infty)$, we have an equivalence $$\lim p'\cong {\operatorname}{Nat}_{K}(\Delta_\ast,p')$$ between the limit of $p'$ and the $\infty$-category of natural transformations from the trivial diagram to $p'$.

In practice, often the best description of diagrams in ${{\mathscr{C}\mathrm{at}}}_\infty$ is in terms of (co)Cartesian fibrations, as developed by Lurie [@Lurie:2009un § 2.4]. Briefly, given an inner fibration between $\infty$-categories ${\mathcal}{C}\to {\mathcal}{D}$ a edge $f:c\to c'$ in ${\mathcal}{C}$ is called *coCartesian* if $${\mathcal}{C}_{f/}\to {\mathcal}{C}_{c/}\times_{{\mathcal}{D}_{p(c)/}}{\mathcal}{D}_{p(f)/}$$ is an equivalence.[^6] The fibration ${\mathcal}{C}\to {\mathcal}{D}$ is called *coCartesian* if there is a coCartesian edge over any edge in ${\mathcal}{D}$ starting at any vertex of ${\mathcal}{C}$. The fibration ${\mathcal}{C}\to{\mathcal}{D}$ is called Cartesian if ${\mathcal}{C}^{op}\to {\mathcal}{D}^{op}$ is coCartesian. Functors $p':K\to {{\mathscr{C}\mathrm{at}}}_\infty$ correspond to coCartesian fibrations $p:X\to K$ via the straightening/unstraightening construction [@Lurie:2009un]; for every $k\in K$, the fibre $X_k:=p^{-1}(k)$ is equivalent to $p'(k)$, and for every edge $k\to k'$ in $K$, the corresponding functor $X_k\to X_{k'}$ is equivalent to one sending any $x\in X_k$ to the target 