The best moment to do so is during the fourth main mission called C2W.
In the last case just look around and maybe you will also listen some barks.
After finding a puppy get close to him and fulton him to the Mother Base.
After that point you should focus on new missions.
When getting close to the 9th or 10th mission you should receive another invitation.
During most missions D-Dog is probably the most useful buddy you’re free to take.
His biggest advantage is the ability to detect the living creatures and important objects from large distance.
It will also help you find the friendly or important characters such as the imprisoned allies.
At first D-Dog may obey some basic orders such as the barking which will draw your enemies' attention.
D-Dog is very efficient while if stays undetected and during one on one combat.
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