In the cross aabb x aabb, what are the genotypes that are homozygous for both traits, and in what proportion will they occur?

In this cross aabb is the only genotype that is homozygous for both traits among 16 offspring's produced during the AaBb × AaBb cross.

Proportion is:   16  :  1  i.e out of 16 only one is homozygous for both traits.

Explanation:

Taking an example of pea plant.

In this question each organism has hetero-zygous  alleles i.e Aa of single character gene with having both dominant and recessive traits.

Say for example in this case capital "A" represents Tall dominant character and small "a" represents dwarf recessive character. Similarly Capital "B" for example  represents round dominant pea seeds and small "b" represents wrinkled recessive pea seeds.

In this question organism with having hetero-zygous characters (i.e having both dominant and recessive traits) are crossed. Underlined represents male part while non-underlined represents Female part of plant

                                                AaBb × AaBb(1)

Four different types of gametes are produced from each organism due to meiosis in which chromosomes become half and alleles become separated.

i.e                     AB,  Ab,  aB and ab × AB,  Ab,  aB and ab(2)

Mating of these alleles are represented below in the form of checker board.

      AB       Ab    aB       ab

AB AABBAABbAaBB  AaBb

Ab AABbAAbbAaBb  Aabb

aB AaBBAaBbaaBB  aaBb

ab AaBbAabbaaBbaabb

In this cross aabb is the only genotype that is homozygous for both traits among 16 offspring's produced during the AaBb × AaBb cross.

Proportion is:   16  :  1