The joining of a sperm and an egg?
The joining is called fertilization. The sperm is said to "fertilize" the egg when it enters the egg, producing a zygote, a one cell embryo. It is now a sperm's DNA paired up with the egg's DNA. Then the zygote starts dividing into more cells and is called a blastocyst (a multi-celled embryo) and when the cells start to differentiate it is called a fetus.
There is no race in the sense of a competition, all sperm are equally affected by the chemical signals. When the sperm reaches the egg, its tail is cast off; thus disposing of male mitochondrial material; and the egg is immediately changed by the entry of the sperm so that another sperm cannot enter. Then follows the joining of the two DNA strands etc.