Wedge over a Functor

Let F : 𝒞 m ⁢ a ⁢ t ⁢ h ⁢ r ⁢ m ⁢ o ⁢ p × 𝒞 → 𝒟 : 𝐹 → superscript 𝒞 𝑚 𝑎 𝑡 ℎ 𝑟 𝑚 𝑜 𝑝 𝒞 𝒟 F:\mathcal{C}^{mathrm{op}}\times\mathcal{C}\to\mathcal{D} be a Functor. A wedge over F 𝐹 F is an object W 𝑊 W an maps w X : W → F ⁢ ( X , X ) : subscript 𝑤 𝑋 → 𝑊 𝐹 𝑋 𝑋 w_{X}:W\to F(X,X) such that for any f : X → Y : 𝑓 → 𝑋 𝑌 f:{{X}\to{Y}} , the following diagram commutes

W𝑊{W}F⁢(X,X)𝐹𝑋𝑋{F(X,X)}F⁢(Y,Y)𝐹𝑌𝑌{F(Y,Y)}F⁢(X,Y)𝐹𝑋𝑌{F(X,Y)}wXsubscript𝑤𝑋\scriptstyle{w_{X}}wYsubscript𝑤𝑌\scriptstyle{w_{Y}}F(mathrmidifx..⋄lsei,f)\scriptstyle{F(mathrm{id}{ifx..lse\par i},f)}F(f,mathrmidifx..⋄lsei)\scriptstyle{F(f,mathrm{id}{ifx..lse\par i})}