A note on Lagrangian intersections and Legendrian Cobordism

Let $\mathrm{\Lambda },{\mathrm{\Lambda }}^{\mathrm{\prime }}$ be a pair of closed Legendrian submanifolds in a closed contact manifold $(Y,\xi =Ker(\alpha ))$ related by a Legendrian cobordism $W\subset (\mathbb{C}\times Y,\tilde{\xi }=Ker(-ydx+\alpha ))$. In this note, we show that in the hypertight setting, if $\mathrm{\Lambda }$ intersects a closed, weakly exact or monotone pre-Lagrangian $P\subset Y$ for reasons of Floer homology, then so does ${\mathrm{\Lambda }}^{\mathrm{\prime }}$.