I work in derived category $D^b(X)$ of constructible sheaves on a reasonable space $X$. Let $j\colon U\to X$ be an open inclusion and $i\colon Y\to X$ the closed complement. Let $M,N\in D^b(X)$ and let $f\in Hom_{D^b(X)}(M,N)$. Suppose $i^*f = 0$ and $j^*f= 0$.

Then is it true that $f=0$?

My gut answer is yes, and I thought I would be able to lever the canonical distinguished triangle $j_! j^* \to id \to i_* i^* \to j_!j^*[1]$ into a proof. But I have failed so far.