Interpolant in a Fibration

Let ℰ → ℬ → ℰ ℬ \mathcal{E}\to\mathcal{B} be a Cartesian Fibration, and let f ⁢ g = h ⁢ k 𝑓 𝑔 ℎ 𝑘 fg=hk be a commuting square in ℰ ℰ \mathcal{E} , as in the following diagram:

I𝐼{I}J𝐽{J}K𝐾{K}L𝐿{L}k𝑘\scriptstyle{k}g𝑔\scriptstyle{g}hℎ\scriptstyle{h}f𝑓\scriptstyle{f}

Now, let A : ℰ J : 𝐴 subscript ℰ 𝐽 A:\mathcal{E}_{J} , and B : ℰ K : 𝐵 subscript ℰ 𝐾 B:\mathcal{E}_{K} , and d : ℰ ⁢ ( g ∗ ⁢ ( A ) , k ∗ ⁢ ( B ) ) : 𝑑 ℰ superscript 𝑔 𝐴 superscript 𝑘 𝐵 d:\mathcal{E}(g^{*}(A),k^{*}(B)) be a vertical morphism in ℰ I subscript ℰ 𝐼 \mathcal{E}_{I} . An interpolant of d 𝑑 d over the square f ⁢ g = h ⁢ k 𝑓 𝑔 ℎ 𝑘 fg=hk consists of:

Such that d = k ∗ ⁢ ( b ) ∘ τ ∘ g ∗ ⁢ ( a ) 𝑑 superscript 𝑘 𝑏 𝜏 superscript 𝑔 𝑎 d=k^{*}(b)\circ\tau\circ g^{*}(a) , where τ 𝜏 \tau is the canonical iso k ∗ ⁢ h ∗ ⁢ ( X ) ≅ g ∗ ⁢ f ∗ ⁢ ( X ) superscript 𝑘 superscript ℎ 𝑋 superscript 𝑔 superscript 𝑓 𝑋 k^{*}h^{*}(X)\cong g^{*}f^{*}(X) induced by the square f ⁢ g = h ⁢ k 𝑓 𝑔 ℎ 𝑘 fg=hk .