Category theory coproduct beginner question I'm reading Jeremy Gibbons's Chapter 5 "Calculating Functional Programs" (online at http://www.cs.ox.ac.uk). He uses some basic category theory, which is new to me. He introduces product and coproduct (printed page numbers 155-6). The discussion of product seems clear to me (my background is some F# programming). But coproduct isn't all the way there for me. Assume the discussion is in the category of sets. 
Here are a few questions representing my trying to comprehend the very basics. Following Gibbons's notation, we have: f::A->C, g::B->C. The constructors, or canonical injections, inl::A->A+B, inr::B->A+B. The morphism "join", f$\bigtriangledown$g::A+B->C. And coproduct sum, or "map", f+g::A,B->C (I think--that's one question, this signature?). 
We have the universal property for join: 
$$h = f\bigtriangledown g \Longleftrightarrow h \circ inl = f \wedge h \circ inr = g $$
And the definition of coproduct sum, or map: 
$$f+g = (inl \circ f)\bigtriangledown (inr \circ g) $$
I'm trying to understand how to apply f$\bigtriangledown$g and f+g. Suppose we have: d | d $\epsilon$ A  $\vee$ d $\epsilon$ B. In order to apply the universal property, it looks to me like we have to apply inl(d) and inr(d). But if d $\epsilon$ A, then what is inr(d), (remember, inr::B->A+B)? Undefined? Empty set? Likewise, in order to apply f+g, it looks to me like we have to apply f(d) and g(d)? And so again, what is g(d) if d $\epsilon$ A, (remember g::B->C)? 
I expect I'm confusing something very simple. I'm at step zero in this material. Thanks for any tips. 
Here's the full link to the text: http://www.cs.ox.ac.uk/jeremy.gibbons/publications/acmmpc-calcfp.pdf
I copied the double colon usage from there. Maybe it's more standard in computer science circles--? If the "A,B->C" is nonsense--that's my fault. I was just trying to guess at it. 
(I'm replying here and not in comments because I can't without reputation.) 
Thanks for both answers. 
Now I see I was very simply misreading f+g when trying to figure out its signature. Let h = inl $\circ$ f, and k = inl $\circ$ g. Then f+g = h $\bigtriangledown$ k. And the signature of $\bigtriangledown$ is from A+B. So I was just out of order and trying to apply f, or g, first (in applying the defintion of f+g). Hopefully I get better...
 A: $\def \codiag {\mathop{\triangledown}}$If we have $f:: A → C$, $g:: B → C$, then $f \codiag g:: A + B → C$. In the category of sets $A + B$ is a disjoint union of $A$ and $B$, and $f \codiag g$ works like this: for $x ∈ A + B$ if $x ∈ A$, then $(f \codiag g)(x) = f(x)$; if $x ∈ B$, then $(f \codiag g)(x) = g(x)$.
If we have $f:: A → C$, $g:: B → D$, then $f + g:: A + B → C + D$. It would be helpful if the text included the information that those injections used in the definition of $f + g$ lead from $cod(f)$ to $cod(f) + cod(g)$ and from $cod(g)$ to $cod(f) + cod(g)$, respectively. So $f + g$ works like $f \codiag g$: if $x ∈ A$ then $(f + g)(x) = f(x)$, but considered as an element of $C + D$ rather than of $C$ (we apply the canonical injection). And similarly if $x ∈ B$.
A: 
Suppose we have: d | d ϵϵ A ∨∨ d ϵϵ B. In order to apply the universal property, it looks to me like we have to apply inl(d) and inr(d). But if d ϵϵ A, then what is inr(d), (remember, inr::B->A+B)? Undefined? Empty set? Likewise, in order to apply f+g, it looks to me like we have to apply f(d) and g(d)?

To talk about elements shared between sets in the category of sets is to talk about their fiber product over some mutual representation, usually $A \cup B$. Since you don't know anything about $A \cup B$, all $d \in A \lor d \in B$ tells you is that $d \in A + B$. So $\operatorname{inr}(d)$ is undefined.
Your definition of $f+g$ doesn't seem to make sense; $f$ and $g$ have codomain $C$ while $\operatorname{inl}$ and $\operatorname{inr}$ have domains $A$ and $B$, so you can't compose them.
A: Hi Jim F1 and welcome to Math.SE. The reference you mentioned is definitely not a good source for learning even the most basic CT (category theory). CT is best done with pictures (diagrams), expecially for beginners. Also note that in the definition of f x g (product bifunctor) on page 155, f and g are not what they are defined in the above paragraph (describing fork). You see this clearly since g cannot compose with exr. Same story goes with f+g.
You should look at Barr's book on CT for Computer Science (free online).
Page 20 describes categories and functional programming.
On page 165 you see the diagram for f x g and on page 178 you see the definition of sum (aka coproduct) and diagram. Unfortunately I cannot draw you a diagram for f+g, but if you take the diagram of f x g on page 165 and change all 3 "x" into "+" and all 4 proj into inl and inr and turn them around, you get precisely the diagram for f + g. 
