Can somebody explain how are they claiming the following highlighted identity in the book Algebraic Theories by Vitale?
The reason I am confused is because the image of the $F$ functor lands inside $\mathcal{B}$ whereas the image of the $\mathcal{B}(-, FX)$ functor lands inside the category whose objects are morphism classes in $\mathcal{B}$.
It certainly is just a misprint. After removing the '$={\cal B}(-,FX)$' part one obtains
'[...] because for $A=Y_{{\cal C}^{op}}(X)={\cal C}(-,X)$, a colimit of $F\cdot\Phi_A$ is $FX$'
which makes perfect sense (cf. Berci's comment)