Let $$ F : \mathcal{C} \to \mathcal{D}, \qquad G : \mathcal{D} \to \mathcal{E}, \qquad H : \mathcal{E} \to \mathcal{F}. $$ For an object $X$, $$ H \circ (G \circ F)(X) = H(G(F(X))). $$ Also $$ (H \circ G) \circ F (X) = H(G(F(X))). $$ Thus the two functors agree on objects. For a morphism $f : X \to Y$, $$ H \circ (G \circ F)(f) = H(G(F(f))). $$ Similarly, $$ (H \circ G) \circ F (f) = H(G(F(f))). $$ Thus the functors agree on morphisms.