You could simple add click event on
dropdown-item
see below
$('.dropdown-menu .dropdown-item').click(function(){
var value = $(this).text();
console.log(value)
// Update the value and show it to the user
$(this).parent().parent().find(".dropdown-toggle").html(value);
});<!DOCTYPE html>
<html lang="en">
<head>
<title>Bootstrap 4 Search Box With Dropdown List</title>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<link rel="stylesheet" href="https://stackpath.bootstrapcdn.com/bootstrap/4.4.1/css/bootstrap.min.css">
<script src="https://code.jquery.com/jquery-3.4.1.slim.min.js"></script>
<script src="https://cdn.jsdelivr.net/npm/popper.js@1.16.0/dist/umd/popper.min.js"></script>
<script src="https://stackpath.bootstrapcdn.com/bootstrap/4.4.1/js/bootstrap.min.js"></script>
<body>
<div class="search-box col-md-5">
<form action="">
<div class="input-group mb-3">
<div class="input-group-prepend">
<button class="btn btn-light dropdown-toggle" type="button" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false">All</button>
<div class="dropdown-menu">
<a class="dropdown-item" href="#">Category One</a>
<a class="dropdown-item" href="#">Category Two</a>
<a class="dropdown-item" href="#">Category Three</a>
<div role="separator" class="dropdown-divider"></div>
<a class="dropdown-item" href="#">Separated Category</a>
</div>
</div>
<input type="text" class="form-control" aria-label="Search input with dropdown button">
<div class="input-group-append">
<button class="btn btn-success" type="button">Search</button>
</div>
</div>
</form>
</div>
</body>
</html>