A functor preserving twisted-arrow limits preserves ends, and dually for coends

By [L1] the wedge $\omega$ over $T$ with vertex $e$ corresponds to the cone $\xi$ over $T\pi$ with ${\xi }_f=T(1_c,f)\circ {\omega }_c$, and $(e,\omega )$ is an end of $T$ exactly when $(e,\xi )$ is a limiting cone; dually the cowedge $\rho$ corresponds to the cocone $\zeta$ under $T{\pi }^{\mathrm{s}\mathrm{w}}$ with ${\zeta }_f={\rho }_c\circ T(f,1_c)$, and $(q,\rho )$ is a coend exactly when $(q,\zeta )$ is a colimiting cocone.

Suppose $F$ preserves $\mathrm{Tw}(\mathcal{C})$-limits. Then $(Fe,F\xi )$ is a limiting cone over $FT\pi$. Functoriality gives $F{\xi }_f=F(T(1_c,f))\circ F{\omega }_c=(FT)(1_c,f)\circ F{\omega }_c$, so $F\xi$ is exactly the cone that [L1] attaches to the family $F\omega$, which is therefore a wedge over $FT$; being limiting, it makes $(Fe,F\omega )$ an end of $FT$.

Suppose $F$ preserves $\mathrm{Tw}(\mathcal{C}{)}^{\mathrm{o}\mathrm{p}}$-colimits. Then $(Fq,F\zeta )$ is a colimiting cocone under $FT{\pi }^{\mathrm{s}\mathrm{w}}$, and $F{\zeta }_f=F{\rho }_c\circ (FT)(f,1_c)$ is the cocone that [L1] attaches to $F\rho$; so $F\rho$ is a cowedge under $FT$ and $(Fq,F\rho )$ is a coend of $FT$.

If $\mathcal{C}$ is small then $\mathrm{Tw}(\mathcal{C})$ is small, since its objects are the morphisms of $\mathcal{C}$ and its morphisms form a subclass of a fourfold product of $\mathrm{Mor}(\mathcal{C})$ with itself; the opposite of a small category is small. So a continuous $F$, which by [F2] preserves all small limits, preserves $\mathrm{Tw}(\mathcal{C})$-limits and step 2.1 applies, and a cocontinuous $F$ preserves $\mathrm{Tw}(\mathcal{C}{)}^{\mathrm{o}\mathrm{p}}$-colimits and step 2.2 applies.