Let $F:\mathcal{A}\to\mathcal{B}$ be a covariant right-exact functor between two abelian categories.
Suppose $\mathcal{A}$ has enough projectives. Then we define the left derived functors of $F$ by $$ L_iF(A)=H_i(F(P_\bullet)) $$ where $A$ is any object in $\mathcal{A}$ and $P_\bullet$ is a projective resolution for $A$ (it can be shown that $L_iF(A)$ is independent of the choice of projective resolution.
Since $F$ is right-exact, the sequence $$ F(P_1)\to F(P_0)\to F(A)\to 0 $$ is exact. Doesn't this mean that $L_0F(A)=H_0(F(P_\bullet)))=0$, since the homology of an exact complex is zero? However everywhere I look says that $L_0F(A)\cong F(A)$.