I've thought about it, and I've decided to keep tabs of the notation used in this "noteblog" in one central location. This is probably incomplete, so apologies.
We write c∈C to indicate c is an object of C; i.e. c∈Ob(C).
If f : c → c' is a morphism in C, we can write f∈homC(c,c'). We can alternatively denote this by f∈C(c,c'). Or, due to my use of random capitalizations, sometimes it may be denoted as f∈HomC(c,c')
Lower case Latin letters starting at a are used for objects, but lower case Latin letters starting in the middle of the alphabet are used for morphisms.
Upper case Latin letters that's bold are used for categories, and unbold upper case Latin letters are used for functors.
Lower case Greek letters are used for natural transformations.
This is probably bad mathematical writing, but "set" will be used in a weaker sense. I mean that "the set of morphisms..." should really be read as "the set (more generally collection) of morphisms...". I assume that most mathematicians would know that a collection is "more general" than a set, so at least I won't have to write so much!
Let C, D be categories. The set (well, more generally collection) of functors from C to D will be denoted by Fun(C, D).
Similarly, if F, G : C → D are functors, the set of natural transformations from F to G is denoted by Nat(F, G).
(The Caveat emptor about random capitalizations also applies to random "lower case-izations", so I may use nat(-,-) for Nat(-,-), or fun(-,-) for Fun(-,-), just be warned I am not trying to pull a fast one! I'm just lazy sometimes...)
Composition of morphisms are read from right to left, like bra-ket notation or Chinese.
Regarding "composition of natural transformations with functors". This is not at first sight mathematically kosher, we can only compose morphisms in the same category and functors are morphisms in Cat, whereas natural transformations are morphisms in functor categories. They are not the same! Do not fret, Wikipedia has the answer:
If η : F ⇒ G is a natural transformation between functors F, G : C → D, and H : D → E is another functor, then we can form the natural transformation Hη : H∘F ⇒ H∘G by defining
(H η)x = Hηx.
If on the other hand K : B → C is a functor, the natural transformation ηK : F∘K ⇒ G∘K is defined by
(η K)x = ηK(x).
Also, I'm not certain if this is standard notation or whether it's something Mac Lane does it Sheaves in Geometry and Logic, but a bijective correspondence between two terms is usually denoted by: We also have derivations, as discussed slightly in §1.2 "Natural Deduction", pp 17ff of my Fascicles 0: Naive Foundations:
Sometimes I will use ▮ to end my proofs, othertimes just "QED" or some other appropriate symbol.