[TYPES] [newbie] is {initial, terminal} object the identity for {coproduct, product}?

[Larry Evans]

Define + as the coproduct operator,
  IOW, X+Y is the coproduct of X and Y for some category C.
Define * as the product operator,
  IOW  X*Y is the product of X and Y for some category C.
Define 0 as the initial object of C.
Define 1 as the terminal object of C.

Is it true that, for all objects, X in C:

  X+0 = X
  0+X = X
  X*1 = X
  1*X = X

?  Also, what's X*0 and X+1?

TIA.

-Larry