I'm fairly confident that the following assertion is true (but I will confess that I did not verify the octahedral axiom yet):
Let $T$ be a triangulated category and $C$ any category (let's say small to avoid alarming my set theorist friends). Then, the category of functors $C \to T$ inherits a natural triangulated structure from T.
By "natural" and "inherits" I mean that the shift map $$ on our functor category sends each $F:C \to T$ to the functor $F$ satisfying $F(c) = F(c)$ on each object $c$ of $C$; and similarly, distinguished triangles of functors $$F \to G \to H \to F$$ are precisely the ones for which over each object $c$ of $C$ we have a distinguished triangle in $T$ of the form $$F(c) \to G(c) \to H(c) \to F(c).$$
The main question is whether this has been written up in some standard book or paper (I couldn't find it in Gelfand-Manin for instance). Perhaps it is considered too obvious and relegated to an elementary exercise. Mostly, I am interested in inheriting t-structures and hearts from $T$ to functor categories $C \to T$, and would appreciate any available reference which deals with such matters.